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We study a two-layer energy balance model that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being partly captured by the other layer, and the effect of all non-radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted εa. The value of εa depends critically on greenhouse gases: increasing concentrations of CO2 and CH4 lead to a more opaque atmosphere with higher values of ϵa. First, we prove that global existence of solutions of the system holds if and only if εa∈(0,2) and blow up in finite time occurs if εa>2. (Note that the physical range of values for εa is (0,1].) Next, we explain the long time dynamics for εa∈(0,2), and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove, in particular, that the surface temperature increases monotonically with respect to εa. This is the key mathematical manifestation of the greenhouse effect., (© 2023 Author(s). Published under an exclusive license by AIP Publishing.)

Many natural systems show emergent phenomena at different scales, leading to scaling regimes with signatures of deterministic chaos at large scales and an apparently random behavior at small scales. These features are usually investigated quantitatively by studying the properties of the underlying attractor, the compact object asymptotically hosting the trajectories of the system with their invariant density in the phase space. This multi-scale nature of natural systems makes it practically impossible to get a clear picture of the attracting set. Indeed, it spans over a wide range of spatial scales and may even change in time due to non-stationary forcing. Here, we combine an adaptive decomposition method with extreme value theory to study the properties of the instantaneous scale-dependent dimension, which has been recently introduced to characterize such temporal and spatial scale-dependent attractors in turbulence and astrophysics. To provide a quantitative analysis of the properties of this metric, we test it on the well-known low-dimensional deterministic Lorenz-63 system perturbed with additive or multiplicative noise. We demonstrate that the properties of the invariant set depend on the scale we are focusing on and that the scale-dependent dimensions can discriminate between additive and multiplicative noise despite the fact that the two cases have exactly the same stationary invariant measure at large scales. The proposed formalism can be generally helpful to investigate the role of multi-scale fluctuations within complex systems, allowing us to deal with the problem of characterizing the role of stochastic fluctuations across a wide range of physical systems.

Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters' value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics' period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the orbit between the neighborhood of the various UPOs. Each UPO and its neighborhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix, we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets.

In this paper, we study phase transitions for weakly interacting multiagent systems. By investigating the linear response of a system composed of a finite number of agents, we are able to probe the emergence in the thermodynamic limit of a singular behavior of the susceptibility. We find clear evidence of the loss of analyticity due to a pole crossing the real axis of frequencies. Such behavior has a degree of universality, as it does not depend on either the applied forcing or on the considered observable. We present results relevant for both equilibrium and nonequilibrium phase transitions by studying the Desai-Zwanzig and Bonilla-Casado-Morillo models.

Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.

Several European countries have suspended the inoculation of the AstraZeneca vaccine out of suspicion that it causes deep vein thrombosis. In this letter, we report some Fermi estimates performed using a stochastic model aimed at making a risk-benefit analysis of the interruption of the delivery of the AstraZeneca vaccine in France and Italy. Our results clearly show that excess deaths due to the interruption of the vaccination campaign injections largely overrun those due to thrombosis even in worst case scenarios of frequency and gravity of the vaccine side effects.

We show that a known condition for having rough basin boundaries in bistable 2D maps holds for high-dimensional bistable systems that possess a unique nonattracting chaotic set embedded in their basin boundaries. The condition for roughness is that the cross-boundary Lyapunov exponent λ x on the nonattracting set is not the maximal one. Furthermore, we provide a formula for the generally noninteger co-dimension of the rough basin boundary, which can be viewed as a generalization of the Kantz-Grassberger formula. This co-dimension that can be at most unity can be thought of as a partial co-dimension, and, so, it can be matched with a Lyapunov exponent. We show in 2D noninvertible- and 3D invertible-minimal models, that, formally, it cannot be matched with λ x . Rather, the partial dimension D 0 (x) that λ x is associated with in the case of rough boundaries is trivially unity. Further results hint that the latter holds also in higher dimensions. This is a peculiar feature of rough fractals. Yet, D 0 (x) cannot be measured via the uncertainty exponent along a line that traverses the boundary. Consequently, one cannot determine whether the boundary is a rough or a filamentary fractal by measuring fractal dimensions. Instead, one needs to measure both the maximal and cross-boundary Lyapunov exponents numerically or experimentally.

Geoengineering can control only some climatic variables but not others, resulting in side-effects. We investigate in an intermediate-complexity climate model the applicability of linear response theory (LRT) to the assessment of a geoengineering method. This application of LRT is twofold. First, our objective (O1) is to assess only the best possible geoengineering scenario by looking for a suitable modulation of solar forcing that can cancel out or otherwise modulate a climate change signal that would result from a rise in carbon dioxide concentration [CO 2 ] alone. Here, we consider only the cancellation of the expected global mean surface air temperature Δ⟨[T s ]⟩. It is in fact a straightforward inverse problem for this solar forcing, and, considering an infinite time period, we use LRT to provide the solution in the frequency domain in closed form as f s (ω)=(Δ⟨[T s ]⟩(ω)-χ g (ω)f g (ω))/χ s (ω), where the χ's are linear susceptibilities. We provide procedures suitable for numerical implementation that apply to finite time periods too. Second, to be able to utilize LRT to quantify side-effects, the response with respect to uncontrolled observables, such as regional averages ⟨T s ⟩, must be approximately linear. Therefore, our objective (O2) here is to assess the linearity of the response. We find that under geoengineering in the sense of (O1), i.e., under combined greenhouse and required solar forcing, the asymptotic response Δ⟨[T s ]⟩ is actually not zero. This turns out not to be due to nonlinearity of the response under geoengineering, but rather a consequence of inaccurate determination of the linear susceptibilities χ. The error is in fact due to a significant quadratic nonlinearity of the response under system identification achieved by a forced experiment. This nonlinear contribution can be easily removed, which results in much better estimates of the linear susceptibility, and, in turn, in a fivefold reduction in Δ⟨[T s ]⟩ under geoengineering practice. This correction dramatically improves also the agreement of the spatial patterns of the predicted linear and the true model responses. However, considering (O2), such an agreement is not perfect and is worse in the case of the precipitation patterns as opposed to surface temperature. Some evidence suggests that it could be due to a greater degree of nonlinearity in the case of precipitation.

Extreme geophysical events are of crucial relevance to our daily life: they threaten human lives and cause property damage. To assess the risk and reduce losses, we need to model and probabilistically predict these events. Parametrizations are computational tools used in the Earth system models, which are aimed at reproducing the impact of unresolved scales on resolved scales. The performance of parametrizations has usually been examined on typical events rather than on extreme events. In this paper, we consider a modified version of the two-level Lorenz'96 model and investigate how two parametrizations of the fast degrees of freedom perform in terms of the representation of extreme events. One parametrization is constructed following Wilks [Q. J. R. Meteorol. Soc. 131, 389-407 (2005)] and is constructed through an empirical fitting procedure; the other parametrization is constructed through the statistical mechanical approach proposed by Wouters and Lucarini [J. Stat. Mech. Theory Exp. 2012, P03003 (2012); J. Stat. Phys. 151, 850-860 (2013)]. The two strategies show different advantages and disadvantages. We discover that the agreement between parametrized models and true model is in general worse when looking at extremes rather than at the bulk of the statistics. The results suggest that stochastic parametrizations should be accurately and specifically tested against their performance on extreme events, as usual optimization procedures might neglect them.

In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and Hènon maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.